Cardinal voting systems: Difference between revisions

[[Arrow's impossibility theorem]] demonstrates the impossibility of designing a deterministic [[ordinal voting]] system which passes a set of desirable criteria. Since Arrow's theorem only applies to [[ordinal voting]] and not cardinal voting systems, several cardinal systems to pass all these criteria. The typical examples are [[score voting]] and [[majority judgment]]. Additionally, there are cardinal systems which fail one of Arrow's criteria, but not due to Arrow's theorem; for example, [[Ebert's method]] fails [[monotonicity]]. Subsequent social choice theorists have expanded on Arrow's central insight, and applied his ideas to specific cardinal systems.

Furthermore, there are other [[Voting paradox| impossibility theorems]] which are different than Arrow's and apply to cardinal systems. The most relevant are Gibbard's theorem and the [[Balinski–Young theorem]].

Gibbard's 1973 theorem holds that any deterministic process of collective decision making with multiple options will have some level of [[strategic voting]].<ref>{{cite journal|last=Gibbard|first=Allan|author-link=Allan Gibbard|year=1973|title=Manipulation of voting schemes: A general result|url=http://www.eecs.harvard.edu/cs286r/courses/fall11/papers/Gibbard73.pdf|journal=Econometrica|volume=41|issue=4|pages=587–601|doi=10.2307/1914083|jstor=1914083}}</ref>. Later results show that even allowing for nondeterminism, only very particular methods are strategy-proof. For example, requiring weak unanimity and assuming voters do not give their utilities with infinite precision, the only strategy-proof cardinal method is random ballot.<ref>{{cite journal | last=Dutta | first=Bhaskar | last2=Peters | first2=Hans | last3=Sen | first3=Arunava | title=Strategy-proof Cardinal Decision Schemes | journal=Social Choice and Welfare | publisher=Springer Science and Business Media LLC | volume=28 | issue=1 | date=2006-05-17 | issn=0176-1714 | doi=10.1007/s00355-006-0152-9 | pages=163–179|url=https://www.researchgate.net/publication/24064783_Strategy-proof_Cardinal_Decision_Schemes}}</ref>

As a result of this much of the work of social choice theorists is to find out what types of [[strategic voting]] a system is susceptible to and the level of susceptibility for each. For example, [[Single-member district|single-member systems]] are not susceptible to [[free riding]]. The [[Balinski–Young theorem]] holds that a system cannot satisfy a [[quota rule]] while being both [[House monotonicity criterion | house monotone]] and [[population monotonicity|population monotone]]. This is important because quota rules are used in most definitions of [[proportional representation]] and [[population monotonicity]] is intimately tied to the [[participation criterion]].